Asymptotic maximization of 1-factors

Prove that the cyclic regular tournament T_n is an asymptotic maximizer for the number of 1-factors among tournaments on n vertices, where a 1-factor is a collection of vertex-disjoint directed cycles covering all vertices.

Background

A 1-factor in a tournament is a spanning collection of vertex-disjoint directed cycles, and every Hamiltonian cycle is a special case of a 1-factor. The paper's methods asymptotically determine the maximum number of Hamiltonian cycles, but they do not determine the corresponding maximum for all spanning cycle covers.

The authors suggest that an adaptation of their permanent and determinant techniques may resolve this broader counting problem, and explicitly formulate the conjecture that the cyclic regular tournament T_n is also asymptotically extremal for 1-factors.